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index i. If /i is Lebesgue measure on R", and if jq = (at t,..., awl-), then we have
(                                ju(P) = |det(av)|.

For let u be the automorphism of the vector space R" given by u(et) = xt for
1 < z <  (where ^,...,  are the vectors of the canonical basis of Rn); we
have P = w(K), where K is the parallelotope on the vectors el9 ...,. Hence
from ( and (14.3.9) we have ju(P) = |det u\ ju(K); but it follows imme-
diately from (13.21.15) that /j,(K) = 1. Hence the formula (

Now let S be the "simplex" on the vectors xl9,.., xn, i.e., the set of all


x =  i*i such that ti ^  for a11 *> and 5i + ' * * + n ^

(                            ...     BI

Here again, by using the same linear transformation w, we are reduced to
proving the formula ( when xt = ei for 1 <* i ^ n. Let Sn denote the
corresponding simplex, write \in in place of ju, and let an = t*n($n)* Identifying
Rn with R""1 x R, consider for each AeR the section Sn(A) in R"""1. This
set is empty if A < 0 or A > 1. If 0 :g A ^ 1, then Sn(A) is the set of points
(1,    > <^-i) e R""1 such that

and is therefore the image of Sn(0) under the homothety with ratio 1  L
Applying ( and (14.3.9) to this homothety in R""1, we obtain

Apply (13.21.8) to the compact set Sn, and we get

Jo                                 n

Since clearly a^ = 1, we have an = l/nl.

(14.3.11)   Application to the calculation of integrals:   II. Closed ball.

With the same notation as in (14.3.10), we shall now calculate the measure
 = nn(Bn) of the Euclidean unit ball Bn, i.e., the set of all (jct, ..., *) e Rn Application to the calculation of integrals:   I. Parallelotope and